Contact information

Research

High performance and parallel computing

In progress.

Two-Phase Flow

In progress.

Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) methods are a powerful class of techniques for approximating solutions to partial differential equations. My interest in DG stems from its strong mathematical theory and robust capabilities in the context of time dependent PDEs.

Classical DG methods give rise to an abundance of degrees of freedom, which poses major challenges for implicit PDEs. The Hybridizable discontinuous Galerkin (HDG) method remedies this by introducing additional Lagrange multiplier unknowns on the element boundaries. With a judicious choice of numerical flux and trace, the HDG method is capable of statically condensing its interior unknowns. This results in a smaller globally coupled linear system in terms of the additional hybrid unknowns only.

A depiction of the Hybridizable discontinuous Galerkin method is given below.

Spectral/Algebraic/Geometric Multigrid

Multigrid methods are very effective multilevel solvers for linear systems. Geometric Multigrid (GMG) takes advantage of a hierarchy of grids or discretizations and reduces the error over a range of frequencies simultaneously. I'm interested in the parallel implementation of GMG applied to linear systems arising from DG discretizations; as well as exporing efficient relaxation, prolongation, and restrication operators.

Spectral Methods

Spectral methods are a class of discretization techniques for partial differential equations that are capable of producing highly accurate approximations. If the underlying function one is approximating is smooth, spectral methods have the fastest possible convergence. See my Master's thesis for an exploration of these methods within the context of hyperbolic PDEs.